Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
New Investigation of Tidal Gravity Results 1 of 16 http://www.upf.pf/ICET/bim/text/ducarme.htm New Investigation of Tidal Gravity Results from the GGP Network Bernard Ducarme*, He-Ping Sun**and Jian-Qiao Xu** *Chercheur Qualifié au FNRS, Observatoire Royal de Belgique Av. Circulaire 3, B-1180 Brussels, email: [email protected] **Institute of Geodesy and Geophysics, Chinese Academy of Sciences 54 Xu Dong road, 430077 Wuhan, China, email: [email protected] Abstract Some 20 series longer than one year are now available in the GGP data bank, including 3 dual sphere instruments. To eliminate the tidal loading effects we interpolated the contribution of the smaller oceanic waves from the 8 well determined ones i.e. Q1, O1, P1, K1, N2, M2, S2 and K2. It was done for six different oceanic models: SCW80, CSR3.0, FES95.2, TPXO2, CSR4.0 and ORI96. In the diurnal band no model is decisively better than the others and a mean tidal loading vector is giving the most stable solution for the study of the liquid core resonance. In the semi-diurnal band however the SCW80 and TPXO2 models are not convenient. We are investigating mainly the diurnal waves around the liquid core resonance i.e. K1, y1 and j1. The scattering of the corrected amplitude factors for the waves O1 and K1 reaches 0.3%. and the tidal factors are determined with a precision slightly better than 0.1%. O1 is fitting perfectly to the DDW99 and MAT01 models but there is an offset of 0.1% for K1. K1 exhibits a slight phase advance with respect to O1. From our data set we computed the FCN period and found values very close to the 429.5 days deduced from the VLBI observations. 1. Introduction Since July 1997 more than 15 superconducting gravimeters are operating in the framework of the Global Geodynamics Project (GGP) (Crossley & al.,1999), following to standardised procedures. Nineteen stations with data sets longer than one year are now available in the GGP data bank, hosted at the Royal Observatory of Belgium, including 3 dual sphere instruments. The one minute sampled original data obtained are pre-processed and analysed at the International Centre for Earth Tides (ICET) using the standard procedures (Ducarme & Vandercoilden, 2000). These data are corrected using a remove restore technique based on the T-soft software (Vauterin, 1998) and decimated to one hour sampling prior to the analysis by ETERNA software (Wenzel, 1996). Atmospheric pressure is the only auxiliary channel available for all the stations. For the dual sphere (CD) instruments the results of the U and L spheres are so close that we computed a common analysis of the two time series. For the stations Strasbourg and Wettzell we consider also the series obtained with the old T005 and T103 instruments. Finally we introduced the results of the renovated ASK228 gravimeter of Pecny (BroZ & al., 1996) which has an RMS error on the unit weight better than many of the oldest cryogenic instruments. Altogether we are thus considering 22 data sets, 10 or them outside Europe. For each of them we are able to extract 22 tidal groups: s1, Q1, r1, O1, NO1, p1, P1, K1, y1, j1, q1, J1, OO1, 2N2, m2, N2, n2, M2, L2, T2, S2, K2. In table 1 we give the details of the different series in increasing ICET station number, which allows to see the regional distribution of stations. For recent instruments, CT or CD series, the RMS error on the unit weight is below one nm.s-2. In Potsdam we have the only T model with an error below 1nms-2. The barometric efficiency is always close to -3nms-2/hPa with three exceptions: the Askania gravimeter at Pecny, a very low coefficient at Sutherland and a very high one at Syowa. For the 19 other series we have as a mean: 3.368 ± 0.036 nms-2/hPa, with a standard deviation s = 0.152 nms-2/hPa. 2/22/2011 9:18 AM New Investigation of Tidal Gravity Results 2 of 16 http://www.upf.pf/ICET/bim/text/ducarme.htm Table 1 Characteristics of the tidal gravity stations T: large dewar, CT: compact, CD : dual sphere, ASK: Askania GGP ICET name Lat. Long. Instr. Data set (days) RMS error Baro. Fact. (nms-2/hPa) BE 0200 Brussels 50.7986 4.3581 T003 6,660 1.743 -3.467±.005 MB 0243 Membach 50.6093 6.0066 CT21 1,728 1.007 -3.286±.006 ST 0306 Strasbourg 48.6223 7.680 T005 3,272 2.265 -3.128±.010 CT26 817 0.797 -3.394±.007 BR 0515 Brasimone 44.1235 11.1183 T015 1,098 2.576 -3.053±.036 VI 0698 Vienna 48.2493 16.3579 CT25 729 0.662 -3.467±.007 WE 0731 Wetzell 49.1458 12.8794 T103 726 2.639 -3.374±.031 CD029 2x291 0.667 -3.340±.009 PO 0765 Potsdam 52.3809 13.0682 T018 2,250 0.855 -3.313±.004 MO 0770 Moxa 50.6450 11.6160 CD34 2x580 0.590 -3.320±.005 ME 0892 Metsahovi 60.2172 24.3958 T020 1614 1.299 -3.636±.007 PC 0930 Pecny 49.9200 14.780 ASK228 412 0.887 -4.894±..013 WU 2647 Wuhan 30.5139 114.4898 CT32 985 0.750 -3.237±..010 KY 2823 Kyoto 35.0278 135.7858 T009 686 3.323 -3.183±..038 MA 2824 Matsushiro 36.5430 138.2070 T011 880 1.163 -3.523±..006 ES 2849 Esashi 39.1511 141.3318 T007 875 1.286 -3.549±..011 SU 3806 Sutherland -32..3814 20.8109 CD37 (491+304) 0.689 -2.657±..013 BA 4100 Bandung -6.8964 107.6317 T008 420 7.450 -3.524±..243 CB 4204 Canberra -35.3206 149.0077 CT31 890 0.776 -3.392±..010 BO 6085 Boulder 40.1308 254.7672 CT024 1,401 0.997 -3.518±..007 CA 6824 Cantley 45.5850 284.1929 T012 2,386 1.443 -3.293±..006 SY 9960 Syowa -69.0070 39.5950 T016 548 1.103 -4.115±.009 2/22/2011 9:18 AM New Investigation of Tidal Gravity Results 3 of 16 http://www.upf.pf/ICET/bim/text/ducarme.htm Table 2 Stability of the tidal analysis results Station Instr. Series d(O1) d(K1) d(M2) d(S2) BE0200 T003 82/86 1.15375 1.14072 1.18404 1.19826 86/91 1.15290 1.13982 1.18334 1.19623 91/96 1.15338 1.13967 1.18410 1.19643 96/00 1.15344 1.14019 1.18379 1.19587 Discr.% @0.1% ME0243 CT21 95/97 1.14939 1.13730 1.18734 1.19285 98/00 1.14932 1.13734 1.18750 1.19225 <0.1% ST0306 T005 CT26 87/91 1.14708 1.13520 1.18496 1.18750 91/96 1.14745 1.13535 1.18510 1.18794 97/99 1.14871 1.13704 1.18657 1.18887 <0.1% @0.1% WE0731 T103 96/98 1.14424 1.13101 1.17973 1.17684 CD29 98/01 1.14820 1.13531 1.18357 1.18170 @0.4% PO0765 T018 92/95 1.15003 1.13732 1.18607 1.18510 95/98 1.14966 1.13738 1.18592 1.18529 <0.1% ME0892 T020 94/96 1.15317 1.14063 1.18178 1.17618 97/00 1.15304 1.14078 1.18090 1.17543 <0.1% CA6824 T012 89/95 1.16655 1.14819 1.20434 1.18479 97/99 1.16530 1.14745 1.20270 1.18270 @0.1% To study the stability of the tidal analysis results we subdivided the longest series on one hand and compared the results of different instruments in the same station on the other hand. From the results of Table 2 it is clear that, in each station, the tidal analysis factors of the main waves are stable at the 0.1% level with the exception of Wettzell. We shall see later on that it is the T103 which is not correctly calibrated. On the contrary at Strasbourg the renovated superconducting gravimeter agrees at the 0.15% level with the older series. As it could be expected the phase differences agree within 0°.5. It should be pointed out that the internal analysis errors are lower by one order of magnitude. 2/22/2011 9:18 AM New Investigation of Tidal Gravity Results 4 of 16 2. http://www.upf.pf/ICET/bim/text/ducarme.htm Efficiency of the tidal loading correction For tidal loading correction we are using 6 different tidal models: SCW80 (Schwiderski, 1980), CSR3.0 (Eanes, 1996), FES95.2 (Le Provost & al., 1994), TPXO2 (Egbert & al., 1994) , CSR4.0 and ORI96 (Matsumoto & al., 1995). Schwiderski is currently used as a working standard since more than 20 years but its coverage is not sufficient in many areas. CSR3 and FES95.2 have been recommended by Shum & al.( 1996) and tested by Melchior & Francis (1996) on tidal gravity data. As a first step we want to compare the efficiency of the different oceanic models for the reduction of the tidal loading influence in our data set. For any tidal wave let us define: - the residual vector B(B,b) expressing the difference between the observed tidal vector A(Aobs,a) and the body tides vector R(ADDW,0) modelled using the Dehant, Defraigne and Wahr (DDW99) non-hydrostatic model (Dehant et al., 1999), with Aobs= Ath.dobs and ADDW= Ath.dDDW , i.e. B=A-R. - the final residual vector X(X,c) expressing the discrepancy between the B vector and the oceanic loading vector L(L,l), i.e. X=B-L. We can also compute their relative importance as B/Ath and X/Ath. On the other hand, for any tidal vector A, we can compute weighted mean vectors (M) for the Diurnal(D) and Semi-Diurnal(SD) families by the formula: MD = [2*A(O1) + A(P1) + 3*A(K1)]/6 MSD = [2*A(M2) +A(S2)]/3 The weight of the waves is proportional to their amplitude. It is justified by the fact that the signal to noise ratio in the tidal gravity recording as well as in the oceanic tidal models is directly proportional to the amplitude of the tidal constituent. In table 4 we express for each station the mean vectors MD,SD (B/Ath) and MD,SD (X/Ath) Table 3. Global comparison of the efficiency of the different oceanic models Diurnal SCW80 % 88.8 CSR3.0 % 93.1 FES95.2 % 96.3 TPX02 % 95.9 CSR4.0 % 94.7 ORI96 % 93.0 emi-Diurnal 82.5 93.6 94.6 77.6 88.5 94.9 MEAN % N 95.8 6 91.7 94.2 6 4 For each wave we can define the efficiency of the tidal loading correction as (B-X)/B. In a similar way the ratios are thus expressing the mean efficiency of the oceanic loading correction for the correspondent tidal family. As a first step we tried to determine a mean efficiency by averaging the results of all the stations for each oceanic model (Table 3). For the diurnal waves efficiencies are larger than 90% except for SCW80. In the semi-diurnal band however two models have a much lower efficiency, SCW80 and TPX02, and it was thus decided to exclude them for the detection of anomalous stations. Detailed investigations showed that it was mainly in Western Europe that these models were less efficient for tidal loading correction. 2/22/2011 9:18 AM New Investigation of Tidal Gravity Results 5 of 16 http://www.upf.pf/ICET/bim/text/ducarme.htm Table 4 . Efficiency ED = (MD(B)-MD(X))/MD(B) and ESD = (MSD(B)-MSD(X))/MSD(B) of tidal loading corrections bands averaged in Diurnal and Semidiurnal bands for different oceanic models Station Diurnal waves, 6 oceanic models No. Name BE0200 Brussels MB0243 Membach ST0306 StrasbourgT StrasbourgC BR0515 Brasimone VI0698 Vienna WE0731 WetzellT WetzellC PO0765 Potsdam MO0770 Moxa ME0892 Metsahovi PC0930 Pecny WU2647 Wuhan KY2823 Kyoto MA2834 Matsushiro ES2849 Esashi SU3806 Sutherland BA4100 Bandung CB4204 Canberra BO6085 Boulder CA6824 Cantley SY9960 Syowa MD(B) Semi-Diurnal waves, 4 oceanic models* MD(X) ED MSD(B) B(%) b(°) X(%) c(°) % B(%) 0.36 0.42 0.46 0.31 0.59 0.43 0.67 0.58 0.37 0.41 0.43 0.29 2.16 5.27 4.77 6.67 0.75 20.88 1.85 3.00 1.59 8.23 93..94 103.19 127.69 111.64 146.84 139.07 158.51 108.94 95.33 111.83 33.50 98.74 -23.39 2.85 4.91 11.98 -57.46 93.94 -61.53 60.87 40.92 5.95 0.39 0.03 0.16 0.11 0.36 0.14 0.48 0.22 0.08 0.03 0.54 011 0.38 0.24 0.19 0.52 0.24 0.57 0.40 0.22 0.49 1.22 -28.54 -55.20 -169.13 -79.75 -157.10 -168.50 -169.57 100.03 22.72 124.34 7.79 -14.71 6.52 57.55 -144.22 -12.96 -74.44 -174.12 -2.57 49.00 -8.42 -18.09 -7.6 91.9 66.3 65.2 39.1 66.9 28.5 62.8 78.7 91.9 -26.0 65.2 82.5 95.5 96.0 92.2 68.5 97.2 78.5 92.6 69.2 85.2 5.10 4.73 4.13 4.14 2.25 2.37 2.68 3.40 3.25 3.34 2.01 2.62 1.38 4.12 3.25 4.63 10.02 2.42 4.64 0.48 3.84 27.08 MSD(X) b(°) X(%) 60.44 56.15 55.51 53.15 47.26 40.45 51.83 52.72 43.80 48.34 30.91 38.76 -27.53 1.88 12.55 33.69 86.57 -37.40 -71.43 77.14 -24.01 0.97 0.50 0.15 0.23 0.29 0.56 0.33 0.59 0.39 0.07 0.19 0.27 0.42 0.29 0.12 0.28 0.63 0.15 0.61 0.33 0.27 0.62 6.36 ESD c(°) % -70.84 -103.95 -120.24 -84.88 -118.43 -120.14 90.3 96.9 94.4 92.9 75.1 86.1 -156.66 78.2 109.45 -57.85 -156.35 -31.26 -105.12 -11.77 104.35 -120.98 -11.005 -29.47 102.27 -84.23 -112.09 -107.37 5.37 88.5 97.7 94.4 86.6 84.1 79.3 97.0 91.3 86.3 98.5 74.6 93.0 43.4 83.7 76.5 * SCW80 and TPX02 excluded For well calibrated instruments large values of MD,SD(X) will correspond to imperfect loading correction. However large values of MD,SD(X) can also points to the stations with calibration errors, if the corresponding phase is close to 0° or 180°. To detect calibration errors we averaged for each station the MD,SD(X) vectors of all the oceanic models, 6 in the diurnal band but only 4 in the semi-diurnal one.. In Table 4 the residues are scaled in function of the theoretical amplitude of the waves and are thus expressed in percentage. In the diurnal band the MD(B) residues are below 0.5% in Europe but reach 5% in Japan and up to 20% in Bandung, due to its very low latitude. In the semi-diurnal band the effect are decreasing from 5% to 2.5% from West to East in Europe. They are generally large elsewhere, except in Boulder. In Syowa and Sutherland the effects to be explained reach more than 10%. Some series are clearly anomalous in both tidal families as Syowa and Bandung. It was expected in Syowa as the station is located at the Antarctic coast and for all the cotidal maps it is covered by water. For Bandung it is well known that Indonesian archipelago has very complex oceanic tides and it is not surprising that the oceanic loading evaluation not precise enough. The calibration is probably not in question as the final 2/22/2011 9:18 AM New Investigation of Tidal Gravity Results 6 of 16 http://www.upf.pf/ICET/bim/text/ducarme.htm residue has a phase of 90°. In Boulder the very low efficiency of the tidal loading correction in SD is simply due to the fact that the loading amplitude is very low in this station. In Europe the calibrations of Brasimone, Brussels and WetzellT are certainly questionable with final residues reaching 0.5%. Outside of Europe Esashi and Cantley seem also offset. For further studies of the liquid core resonance effects we can conclude that, as previously noticed by Melchior & Francis (1996), no model is decisively better than the others in the diurnal band (Table 3) and that a mean tidal loading vector will probably give the most stable solution. The stations of Syowa and Bandung are probably to be rejected. A calibration error close to 0.5% is strongly suspected in Brussels(BE), Brasimone(BR), Cantley(CA), Esashi(ES) and WetzellT (WE,T103) and perhaps in Metsahovi(ME). Similar conclusions have been expressed by Baker & Bos (2001) using the FES99 oceanic model 3. Interpolated tidal loading effects The tidal load vectors are directly proportional to the amplitude of the wave in the exciting tidal potential and the change of phase exhibits a regular behaviour with respect to the frequency shift. Figure 1 shows typical examples of the frequency dependant phase shift of the load vectors for different parts of the world. Therefore it is rather easy to interpolate the load vectors for the smaller components starting from the eight major components (Q1, O1, P1, K1, N2, M2, S2, K2). The load vectors have to be first normalised dividing by their theoretical amplitude in the tidal potential. However in the Diurnal band the most interesting weak components y1 and j1 need in fact to be extrapolated as they are outside of the frequency band extending from Q1 to K1. Another difficulty is that the core resonance does also affect the oceanic tides ( Wahr and Sasao, 1981). To assume a smooth behaviour we have thus first to correct this resonance effect on the main diurnal waves, especially K1, interpolate or extrapolate the weaker components and apply again the resonance on the results. The effect of the resonance is clearly seen on Figure 1. This procedure has been applied on the real and imaginary parts of the oceanic load vectors computed using the 6 selected oceanic models. We computed 14 additional components: s1, r1, NO1, p1, y1, j1, q1, J1, OO1, 2N2, m2, n2, L2 and T2. The efficiency of the ocean load corrections observed for the main tidal waves is confirmed for the weaker constituents which have been interpolated or even extrapolated. In the diurnal band the mean amplitude factor is uniformly reduced of 0.6% to 0.7% for all the waves and we are thus confident in the fact that our tidal loading correction is also improving the results of the small resonant waves y1 and j1. 2/22/2011 9:18 AM New Investigation of Tidal Gravity Results 7 of 16 http://www.upf.pf/ICET/bim/text/ducarme.htm Figure 1. Phase shift of the oceanic load vectors(Diurnal band) STR: Strasbourg, WUH: Wuhan, CAB: Canberra, BOU:Boulder, CAT: Cantley 4. Mean corrected tidal factors in the diurnal band For each of the 13 diurnal wave groups we computed the corrected tidal gravity vectors Ac(dcAth, ac) using the 6 tidal models, by the relation: Ac(dcAth, ac) = A(Aobs,a) – L(L,l) Our main goal was to compute the best "mean" tidal factors for each wave group by averaging the six tidal models first and then the 22 series of observations. For each wave we constructed a double entry table similar to Table 5 and computed the mean value for each line or series. At this level it is important to reject the anomalous stations. In paragraph 2 we found several anomalous data sets. The problem is to know if they should be rejected beforehand. For station with suspicious calibration at the 0.5% level it is worth to point out that their effect is practically cancelling on the mean as we find three station with too low values( Brasimone, WetzellT and Bandung) and three with too high values (Brussels, Metsahovi and Esashi). Bandung is off set by more than 1% and has certainly to be rejected. On the other hand phase anomalies are also an important criterion to reject series. Moreover for the smaller constituents the noise level becomes a larger source of variability than an 0.5% calibration error, especially for stations with a large RMS error (Table 1). We decided thus to apply simply the criterion of the 3s level rejection on amplitude factors and phase differences for the 22 series. As expected Syowa was always rejected. Bandung was accepted for the largest constituents (K1, O1 and P1). Kyoto was bad for all the small waves i.e. waves with amplitude lower than 2µgal at 45° or 5% of K1. Brasimone was rejected 6 times and a few other stations accidentally. For P1 and K1 we kept 21 data sets and as a minimum 17 for y1. The lowest standard deviations are 0.3% on the amplitude factors and 0.08° on the phase differences for P1 and K1. It explains why only stations with a large calibration error will be rejected as for example Syowa in Table 7. For elimination due to the phase we see two examples in Table 2/22/2011 9:18 AM New Investigation of Tidal Gravity Results 8 of 16 http://www.upf.pf/ICET/bim/text/ducarme.htm 5: Syowa and Wetzell C. We can draw some interesting conclusions from Table 5. As already stated the mean observed tidal factor 1.1614 is reduced to 1.1544 and the discrepancy with the DDW99 model is thus reduced from 0.62% to 0.1%. We see also that the standard deviations on the lines are systematically different from the standard deviations on the columns . We computed the standard deviation of the mean corrected amplitude factor and phase difference (dc = 1.1544, ac = 0.01°) starting either from each series averaged over the 6 maps (column MEAN1) s1 or from the average of the 20 series for each map (line MEAN2) s2 and found very different results: on dc s1 = 0.32% and s2 = 0.07% on ac s1 = 0.045° and s2 = 0.025° . If we consider that the variance on each element of the table is the same, the variance on the mean is divided by the number of elements i.e. 6 for a line or 20 for a column. The two standard deviations should then be in the ratio of the square roots i.e. 1.8. It is true for the phase differences with a ratio s1 /s2 = 1.8, but it is not true for the amplitude factor with a ratio 4.6 i.e. 2.5 times larger. On one hand it means that the dispersion due to the calibration errors in the stations is larger than the dispersion due the difference between the oceanic tidal models. On the other hand it confirms that the phase lag corrections are correct at the level of a few seconds. In Table 6 we compare s1 and s2.(ÖN/6) and we see that this noise amplification for the d factors is true for all the waves except at the very edge of the spectrum for the waves s1 and OO1, where the extrapolation of the oceanic tides models increases the noise. 2/22/2011 9:18 AM New Investigation of Tidal Gravity Results 9 of 16 http://www.upf.pf/ICET/bim/text/ducarme.htm Table 5 Complete data set for wave O1 Data sets *SY9960 ME0892 PO0765 BE0200 MO0770 MB0243 PC0930 WE0731T *WE0731C ST0306T ST0306C VI0698 CA6824 BR0515 BO6085 ES2849 MA2834 CB4204 Observed d a 1.2690 .84 1.1531 .25 1.1499 .19 1.1533 .07 1.1487 .14 1.1494 .11 1.1489 .08 1.1442 .12 1.1482 .23 1.1472 .07 1.1487 .06 1.1479 .10 1.1663 .54 1.1462 .07 1.1642 1.31 1.2215 1.33 1.2043 .70 1.1748 -.76 KY2823 SU3806 WU2647 BA4100 1.2097 .59 1.1636 .12 1.1793 -.38 1.1254 10.93 MEAN 2 STD.DEV. 1.1614 Stations are ordered according to absolute value of latitude SCW80 CSR3.0 FES96.2 TPXO2 d a d a d a d a 1.1814 -.11 1.1698 -.83 1.1625 .11 1.1685 .45 1.1567 .03 1.1570 .10 1.1566 .05 1.1577 .07 1.1537 .09 1.1543 .10 1.1536 .07 1.1547 .13 1.1566 .01 1.1577 .04 1.1569 -.02 1.1582 -.03 1.1527 .06 1.1534 .09 1.1525 .04 1.1534 .10 1.1530 .05 1.1541 .08 1.1533 .02 1.1541 .06 1.1526 .01 1.1535 .04 1.1524 -.01 1.1535 .03 1.1480 .05 1.1489 .08 1.1479 .03 1.1488 .07 1.1520 .17 1.1528 .19 1.1518 .14 1.1528 .19 1.1515 .04 1.1524 .06 1.1513 .01 1.1522 .05 1.1530 .02 1.1539 .04 1.1528 -.01 1.1537 .03 1.1514 .03 1.1525 .05 1.1513 .00 1.1524 .05 1.1590 -.01 1.1579 .05 1.1587 -.03 1.1590 -.05 1.1500 .06 1.1508 -.01 1.1498 -.07 1.1496 .02 1.1552 .15 1.1537 .08 1.1563 .02 1.1551 .02 1.1540 -.09 1.1605 -.12 1.1623 .12 1.1619 .17 1.1486 -.13 1.1535 -.18 1.1535 .03 1.1542 .11 1.1593 -.14 1.1566 -.02 1.1541 -.16 1.157 .07 1.1523 -.01 1.1555 -.11 1.1571 -.03 1.1587 *1.07 .010 1.1537 0.0032 1.1563 -.06 1.1558 .15 1.1543 .20 1.1535 -.16 1.1533 -.11 1.1554 .01 1.1590 -.15 1.1583 .08 1.1584 .14 1.1586 .67 1.1460 .02 1.1528 -.66 CSR4.0 d a 1.1596 -.87 1.1591 .03 1.1544 .08 1.1574 -.04 1.1534 .05 1.1539 .02 1.1534 -.02 1.1488 .03 1.1528 .15 1.1521 .01 1.1536 -.01 1.1524 .00 1.1579 .01 1.1504 -.05 1.1542 .04 1.1580 -.06 1.1529 .00 1.1567 .00 ORI96 a d 1.1698 -. 1.1653 . 1.1547 . 1.1577 -. 1.1535 -. 1.1539 -. 1.1537 -. 1.1490 -. 1.1530 . 1.1523 -. 1.1538 -. 1.1524 -. 1.1586 -. 1.1506 -. 1.1547 . 1.1612 -. 1.1541 -. 1.1575 . 1.1547 .06 1.1567 -. 1.1533 -.10 1.1543 -. 1.1586 -.05 1.1588 -. 1.1394 .16 1.1413 *1. .004 1.1549 .039 1.1538 .012 1.1549 .030 1.1537 .008 1.1554 -.0 .076 0.0029 .175 0.0038 .071 0.0032 .174 0.0043 .057 0.0038 .0 stations to be eliminated ampl. phase 1 SY09960* SY09960* 2 WE07313* 2/22/2011 9:18 AM New Investigation of Tidal Gravity Results 10 of 16 http://www.upf.pf/ICET/bim/text/ducarme.htm 5. Comparison of the experimental results with different models In table 6 we give the mean corrected tidal factors and phase differences for the 13 diurnal components and we compare them with several models DDW99 (non hydrostatic), MAT01 (Mathews, 2001) and several models obtained by fitting the data on a resonance model, using the waves O1, P1, K1, y1 and f1 i.e.: SDX1, SDX2 and SDX3 (cfr §6 and Table 7). The models are normalised on O1 with d = 1.1544 and a = 0.0°. In Table 6 we see that in fact these three models are very close when expressed in terms of amplitude factors and phase differences. On K1 the difference is only at the level of the fourth decimal, smaller than the associated RMS error on the tidal factor. Even on y1 the difference is at the level of the experimental errors. As expected the SXD2 and SXD3 models, which are using directly the experimental values of Table 8, fit very well y1, with a slight positive phase For what concerns the comparison of the mean observed tidal factors with the DDW99 (non hydrostatic) and MAT01 models there are some contradictory remarks. These models are perfectly fitting O1 at the level of the RMS errors, but are 0.1% too low with respect to K1. In a previous reduction with 15 series(Ducarme & Sun, 2001) we already found a similar offset with d(K1) = 1.1356. Concerning the slight phase advance of K1 with respect to O1, forecasted by Mathews, no firm conclusion is possible as its magnitude is of the order of the associated RMS error on the phase differences. The positive result obtained in Ducarme & Sun (2001) is probably an artefact of the Schwiderski model in Western Europe. Concerning the small resonant waves y1,and j1 the RMS errors have been largely improved as, in the previous paper, we had only 9 selected stations, most of them in Europe. The model of Mathews fits better the observations and the experimental models, especially for j1. 6. The Free Core Nutation As already explained, we computed different solutions for the Free Core Nutation (FCN) deduced from our experimental data. SDX1: stacking of the results of 19 stations (Sun & al., 2002) SDX2: direct computation from the experimental results of table 8 with a weight inversely proportional to the RMS error on the amplitude factors s1/ÖN SDX3: direct computation from the experimental results of table 8 with a weight proportional to Ath.ÖN/s1 The best solutions are obtained with the mean of the oceanic models (6 maps). The three solutions converge to the value deduced from the VLBI observations: 429.5 days. It should be pointed out that the Mathews model is associated with a FCN period of 430,04 (429.93-430.48) days (Mathews & al.,2002). In the DDW99 model (Dehant & al., 1999) the FCN period was forced on 431 days, which was the value given by the VLBI observations at that time. For SDX2 and SDX3 we also performed individual computations with the results obtained using each of the 6 oceanic models for loading corrections. These solutions are scattered between 425.3 and 435.6. CSR4.0 gives the minimum value and TPXO2 the maximum one. ORI96 provides also a too large value. The two approaches SXD2 and SXD3 are always very close. 2/22/2011 9:18 AM New Investigation of Tidal Gravity Results 11 of 16 http://www.upf.pf/ICET/bim/text/ducarme.htm Table 6 Mean tidal factors for the diurnal waves Mean factors s1 s2* Wave dc ac % % N (ed) (ea) s1 1.1550 0.164 19 ±.0011 ±.043 Q1 1.1538 0.044 20 ±.0008 ±.026 r1 1.1545 0.017 17 ±.0009 ±.048 ·O1 1.1544 0.010 20 ±.0007 ±.010 NO1 1.1553 -0.023 19 ±.0012 ±.041 p1 1.1510 -0.054 17 ±.0016 ±.091 P1 1.1501 -0.039 21 ±.0006 ±.016 K1 1.1362 0.025 21 ±.0007 ±.015 y1 1.2630 0.073 16 ±.0034 ±.230 j1 1.1691 0.061 19 ±.0018 ±.096 q1 1.1569 0.097 18 ±.0019 ±.132 J1 1.1557 0.014 19 ±.0012 ±.055 OO1 1.1513 0.300 18 ±.0017 ±.071 0.47 0.37 0.35 0.32 0.58 0.67 0.29 0.31 1.36 0.79 0.79 0.52 0.75 0.52 0.25 0.20 0.13 0.20 0.12 0.13 0.11 0.10 0.25 0.24 0.27 0.58 DDW 1.1542 1.1543 1.1543 1.1543 1.1539 1.1507 1.1491 1.1348 1.2717 1.1706 1.1571 1.1569 1.1563 MAT01 SXD1 SXD2 SXD3 d d d d (a) (a) (a) (a) 1.1541 1.15467 1.15467 1.15467 -0.028 0.000 0.000 0.000 1.1541 1.15458 1.15459 1.15459 -0.026 0.000 0.000 0.000 1.1541 1.15457 1.15457 1.15457 -0.026 0.000 0.000 0.000 1.1540 1.15440 1.15440 1.15440 -0.024 0.000 0.000 0.000 1.1535 1.15386 1.15385 1.15385 -0.021 0.000 0.000 0.000 1.1504 1.15091 1.15087 1.15087 -0.008 -0.002 -0.002 0.000 1.1489 1.14949 1.14953 1.14942 -0.002 -0.003 -0.003 0.000 1.1349 1.13664 1.13643 1.13641 0.062 -0.004 -0.012 0.004 1.2655 1.25993 1.26302 1.26217 0.022 0.360 0.071 0.110 1.1693 1.16856 1.16878 1.16876 -0.068 0.016 0.010 0.002 1.1564 1.15643 1.15646 1.15646 -0.028 0.002 0.001 0.000 1.1562 1.15622 1.15625 1.15625 -0.027 0.001 0.001 0.000 1.1556 1.15557 1.15559 1.15559 -0.024 0.001 0.001 0.00 N: number of series, s1: standard deviation on series, s2*= s2.(ÖN/6) : normalised standard deviation on 2/22/2011 9:18 AM New Investigation of Tidal Gravity Results 12 of 16 http://www.upf.pf/ICET/bim/text/ducarme.htm oceanic models, e : RMS errors on tidal factors, · reference for SXD models. 2/22/2011 9:18 AM New Investigation of Tidal Gravity Results 13 of 16 http://www.upf.pf/ICET/bim/text/ducarme.htm Table 7 Experimental models of the core resonance Oceanic model SXD1 T(days) SXD2 Q T(days) SXD3 Q T(days) Q 6 maps 429.9 20769 429.1 -1725650 429.7 54871 SCW80 432.1 12760 428.3 280847 429.5 78378 CSR3.0 428.6 28503 429.1 -34855 429.3 -75594 FES95.2 432.9 17492 427.9 1467138 427.2 41093 TPXO2 425.9 16250 432.4 -31252 435.6 -117705 CSR4.0 434.7 -60940 425.9 1732741 425.3 43722 ORI96 434.6 9387 431.8 24732 433.3 15152 SXD: Sun-Xu-Ducarme (Sun & al., 2002b) 7. The semi-diurnal waves We applied a similar procedure to compute mean tidal factors and phase differences for the 9 semi-diurnal components (Table 8).It seems that the extrapolation of the oceanic loading is less stable than in the diurnal band. Very large error bars are associated with 2N2 and µ2, T2 and n2 give satisfactory results but not L2. In table 8 we give the solutions using 6 and 4 tidal models respectively. The systematic difference of the results obtained using SCW80 and TPX02, already detected in §2, is fully confirmed. The standard deviation s2 is greatly reduced using only 4 models. For all the waves except K2 there is a systematic increase of at least 0.1% after the suppression of the 2 models. For M2 the mean corrected tidal factors computed using SCW80 and TPX02 are 0.5% lower than the values obtained with the 4 other ones, which agree between themselves to within 0.05% The solution with 4 oceanic models agree with the reference value 1.1619 for the two major constituents M2 and S2. For what concerns the phase differences they are close to zero for the frequency band ranging from N2 to M2, but there is a large systematic offset of -0.25° from L2 to S2. It seems that the more turbulent characteristics of the semi-diurnal oceanic tides are responsible of the less stable solutions and increase the difference between the oceanic models. We should investigate more recent models such as FES99. 2/22/2011 9:18 AM New Investigation of Tidal Gravity Results 14 of 16 http://www.upf.pf/ICET/bim/text/ducarme.htm Table 8 Averaged tidal factors for the semi-diurnal waves Mean factors s1 s2* DDW Mean factors 6 maps s1 s2** % % 4maps Wave dc ac N (ed) (ea) 2N2 1.1623 -0.169 13 ±.0019 ±.132 m2 1.1609 -0.247 19 ±.0018 ±.0245 N2 1.1613 -0.086 18 ±.0009 ±.048 n2 1.1598 -0.045 21 ±.0011 ±.035 M2 1.1602 0.008 19 ±.0007 ±.021 L2 1.1604 -0.313 16 ±.0029 ±.099 T2 1.1616 -0.360 15 ±.0013 ±.061 S2 1.1603 -0.229 19 ±.0004 ±.032 K2 1.1634 -0.034 20 ±.0007 ±.047 % 0.68 0.80 0.35 0.50 0.31 1.14 0.51 0.16 0.31 % 0.84 1.11 0.81 0.80 0.55 0.31 0.35 0.34 0.40 1.1619 1.1619 1.1619 1.1619 1.1619 1.1619 1.1619 1.1619 1.1619 dc ac (ec) (ec) 1.1658 -0.247 ±-.0023 ±.166 1.1646 -0.306 ±.0024 ±.155 1.1631 -0.110 ±.0009 ±.033 1.1618 -0.036 ±0.009 ±.0034 1.1621 -0.009 ±.0006 ±.024 1.1615 -0.297 ±.0030 ±.0092 1.1625 -0.432 ±.0012 ±.066 1.1614 -0.280 ±0.004 ±0.037 1.1638 -0.114 ±.0008 ±.060 0.83 0.20 1.06 0.26 0.41 0.11 0.40 0.16 0.28 0.11 1.18 0.16 0.48 0.39 0.20 0.17 0.35 0.38 N: number of series, s1: standard deviation on series, s2*= s2.(ÖN/6) : normalised standard deviation (6 oceanic models), s2**= s2.(ÖN/4) : normalised standard deviation (4 oceanic models), e : RMS errors on tidal factors. 8. Conclusions We performed a careful study of the results obtained with superconducting gravimeters installed in 20 different stations, most of them belonging to the GGP network (Crossley et al., 1999). To take full advantage from the unprecedented precision of these data, it was necessary to compute tidal loading corrections not only for eight main waves (Q1, O1, P1, K1, N2, M2, S2, K2) but also for other weaker components, especially in the diurnal band near the liquid core resonance frequency. We had thus to interpolate or extrapolate the existing load vectors at neighbouring frequencies, taking into account the effect of the resonance itself on the 2/22/2011 9:18 AM New Investigation of Tidal Gravity Results 15 of 16 http://www.upf.pf/ICET/bim/text/ducarme.htm oceanic tides, in order to keep only smooth functions of the frequency. We used 6 different tidal models SCW80, CSR3.0, FES96.2, TPXO2, CSR4.0 and ORI96. To have an idea of the efficiency of these load corrections, we used weighted average of the main diurnal or semi-diurnal components to compute the discrepancy between the observed tidal vectors and the DDW99 non-hydrostatic model, before and after tidal loading correction. We found efficiencies close to 95% in the diurnal as well as the semi-diurnal band for the mean of the 6 tidal models. However SCW80 is less efficient in the two tidal bands and TPXO2 not convenient in the semi-diurnal one. We found that there are still calibration errors at the 0.5% level in the GGP network. This fact is emphasised by the fact that, for most of the tidal waves, the standard deviation on the stations is 2.5 times larger than the standard deviation on the oceanic models. We computed for 13 diurnal components mean values of the corrected amplitude factors and phase differences and compared them with several models of the FCN resonance. The mean amplitude factor for O1 agrees perfectly with the DDW99 non-hydrostatic model as well as with the MAT01 model, but there is an offset of 0.1% on K1. The model of Mathews fits better the observed resonance. However we cannot confirm the predictions of the MAT01 model i.e. a slight phase advance for K1 and y1,contrasting with a phase lag for j1 as the RMS error on the phases is still too large. A phase very slight phase advance of K1 is also found in the SXD3 model. Different computations of the FCN period from our data set gave quite satisfactory results as we obtain periods comprised between 429.1 and 429.9 days, converging towards the value 429.5 days deduced from VLBI observations. For the 9 semi-diurnal constituents the modelling is not good. SCW80 and TPX02 give too low corrected amplitudes factors (0.5% for M2) and the interpolation procedure is not stable. It seems that the more turbulent characteristics of the semi-diurnal oceanic tides are responsible of less stable solutions and increase the differences between oceanic models. Acknowledgements The authors are grateful to D.Crossley , GGP Chairman, and all the GGP station managers: G.Casula (Brasimone, I), H.J.Dittfeld (Potsdam, D), M.&G.Harnisch (Wettzell, D), J.Hinderer (Strasbourg, F), Y.Imanishi(Matsushiro,JP), C.Kroner(Moxa,D), J.Merriam (Cantley, CDN), B.Meurers (Vienna,A), J.Neumeyer ( Sutherland, SA), T.Sato(Esashi, JP and Canberra, AUS), K.Shibuya (Syowa, Antarctica), S.Takemoto (Bandung, Indonesia and Kyoto, JP), M.Van Camp (Membach, B), T.Van Dam and D.Crossley (Boulder, USA) and H.Virtanen (Metsahovi, SF). Z.Simon provided the excellent data set of Pecny, Czech Republic. The authors wishes to express their thanks to M.Hendrickx and L.Vandercoilden, from the Royal Observatory of Belgium, who are maintaining the GGP data base at ICET and preprocessing all the incoming data before decimation to hourly intervals. This study was realised in the framework of the Bilateral Scientific and Technical Agreements between Belgium and China ( project "SG observations and Geodynamics", BL/33/C17). H.P.Sun and J.Q. Xu are partly supported by the Natural Sciences Foundation of China, grants no 49925411 and 40174022. References Baker, T.F., Bos, M.S. (2001): Tidal gravity observations and ocean tide models. Proc. 14th Int. Symp. on Earth Tides, Journal of the Geodetic Society of Japan, 47, 1, 76-81. Broz, J., Simon Z., Zeman, A.(1997): Tidal station Pecny: Results of 20 years of observation with the gravimeters GS15 No 228. VUGTK, Proceedings of Research Work 1996, 75-88 Crossley, D., Hinderer, J., Casula G., Francis O., Hsu, H.T., Imanishi, Y., Jentzsch G., Kääriäinen J., Merriam, J., Meurers B., Neumeyer J., Richter B., Shibuya K., Sato T., Van Dam T. (1999): Network of superconducting gravimeters benefits a number of disciplines. EOS, 80, 11, 121/125-126 2/22/2011 9:18 AM New Investigation of Tidal Gravity Results 16 of 16 Dehant, V., Defraigne, P. and Wahr, J. (1999): Tides for a convective Earth http://www.upf.pf/ICET/bim/text/ducarme.htm J. Geophys. Res., 104, B1, 1035-1058. Ducarme, B., Sun, H.P. (2001): Tidal gravity results from GGP network in connection with tidal loading and Earth response. Proc. 14th Int. Symp. on Earth Tides, Journal of the Geodetic Society of Japan, 47, 1, 308-315. Ducarme, B., Vandercoilden, L. (2000): First results of the GGP data bank at ICET. Proc. Workshop "High Precision Gravity Measurements with application to Geodynamics, March 24-26,1999. Cahiers du Centre Européen de Géodynamique et de Séismologie, 17, 117-124. Eanes, R., Bettadpur (1996): The CSR3.0 global ocean tide model: Diurnal and Semi-diurnal ocean tides from TOPEX/POSEIDON altimetry, CRS-TM-96-05, Univ. of Texas, Centre for Space Research, Austin, Texas Egbert, G., Bennett, A., Foreman, M. (1994): TOPEX/Poseidon tides estimated using a global inverse model. Journal of Geophysical Research, 99(C12), 24821-24852. Le Provost, C., Genco, M.L., Lyard, F., Vincent, P., Canceil, P.(1994): Spectroscopy of the ocean tides from a finite element hydrodynamic model. Journal of Geophysical Research, 99(C12), 24777-24797. Mathews, P.M. (2001): Love numbers and gravimetric factor for diurnal tides. Proc. 14th Int. Symp. on Earth Tides, Journal of the Geodetic Society of Japan, 47, 1, 231-236. .Mathews, P.M., Herring, T.A., Buffett, B.A. (2002): Modeling of nutation-precession: New nutation series for nonrigid Earth and insights into the Earth's interior. Journal of Geophysical Research (under press). Matsumoto, K., Ooe, M., Sato, T., Segawa, J. (1995): Ocean tides model obtained from TOPEX/POSEIDON altimeter data. J. Geophys Res., 100, 25319-25330. Melchior, P., Francis, O. (1996): Comparison of recent ocean tide models using ground-based tidal gravity measurements. Marine Geodesy, 19, 291-330 Schwiderski, E.W. (1980): Ocean Tides I, Global ocean tidal equations. Marine Geodesy, 3, 161-217. Shum, C.K., Andersen, O.B., Egbert, G., Francis, O., King, C., Klosko, S., Le Provost, C., Li X., Molines, J.M., Parke, M., Ray, R., Schlax, M., Stammer D., Temey, C., Vincent P., Woodworth P.L., Wunsch C.(1996): Comparison of newly available deep ocean tide models by the TOPEX/POSEIDON Science Working Team. Journal of Geophysical Research Sun, H.P., Ducarme, B., Xu, J.Q.(2002): Determination of the Free Core Nutation Parameters by Stacking Tidal Gravity Measurements from GGP network. Proc. GGP Workshop, Jena, March 11-15,2002. Vauterin, P. (1998): Tsoft: graphical & interactive software for the analysis of Earth Tide data, Proc. Thirteenth Int. Symp. Earth Tides, Brussels, July 22-25 1997, Observatoire Royal de Belgique, Série Géophysique, 481- 486. Wahr, J.M., Sasao, T. (1981): A diurnal resonance in the ocean tide and in the Earth's load response due to the resonant free core nutation. Geoph. J. R. Astr. Soc., 64, 747-765. Wenzel, H.G.(1996): The nanogal software: data processing package ETERNA 3.3, Bull. Inf. Marées Terrestres, 124, 9425-9439. 2/22/2011 9:18 AM